\documentclass{article}
\begin{document}
This is a simple math example:$$c^2 = a^2 + b^2$$
\begin{equation}
  \label{eq:1}
  c^2 = a^2 + b^2
\end{equation}
\begin{equation}
  \label{eq:aa}
  num = 123 + 789
\end{equation}

common math\\
$a_1$ \qquad $x^{2}$ \qquad
$e^{-\alpha t}$ \qquad
$a^{3}_{ij}$\\
$e^{x^2} \neq {e^x}^2$\\

square root:\\

$\sqrt{x}$ \qquad
$\sqrt{x^{2} + \sqrt{y} }$
\qquad $\sqrt[3]{2}$\\[3pt]
$\surd[x^2 + y^2]$\\

overline and underline: \\
$\overline{m + n}$ \qquad
$\underline{m + n}$

overbrace and ubderbrace; \\
$$\underbrace{ a + b + \cdots + z }_{26}$$\\

vectors: \\
\begin{displaymath}
  \vec a \qquad\overrightarrow{AB}
\end{displaymath}

fraction:\\
$1\frac{1}{2}$~hours
\begin{displaymath}
  \frac{ x^{2} }{k + 1} \qquad
  x^{\frac{2}{k + 1} } \qquad
  x^{ 1/2 }
\end{displaymath}

integral operator & sum operator & product operator: \\
\begin{displaymath}
  \sum_{i = 1}^{n} \qquad
  \int_{0}^{\frac{\pi}{2}} \qquad
  \prod_\epsilon
\end{displaymath}
\end{document}